## Measure and Integration: A Concise Introduction to Real Analysis (Google eBook)A uniquely accessible book for general measure and integration, emphasizing the real line, Euclidean space, and the underlying role of translation in real analysis
The author develops the theory of measure and integration on abstract measure spaces with an emphasis of the real line and Euclidean space. Additional topical coverage includes: Measure spaces, outer measures, and extension theorems - Lebesgue measure on the line and in Euclidean space
- Measurable functions, Egoroff's theorem, and Lusin's theorem
- Convergence theorems for integrals
- Product measures and Fubini's theorem
- Differentiation theorems for functions of real variables
- Decomposition theorems for signed measures
- Absolute continuity and the Radon-Nikodym theorem
- Lp spaces, continuous-function spaces, and duality theorems
- Translation-invariant subspaces of L2 and applications
The book's presentation lays the foundation for further study of functional analysis, harmonic analysis, and probability, and its treatment of real analysis highlights the fundamental role of translations. Each theorem is accompanied by opportunities to employ the concept, as numerous exercises explore applications including convolutions, Fourier transforms, and differentiation across the integral sign. Providing an efficient and readable treatment of this classical subject, |

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### Contents

1 | |

11 | |

A Concise Introduction to Real Analysis 3 Lebesgue Measure | 31 |

A Concise Introduction to Real Analysis 4 Measurable Functions | 55 |

A Concise Introduction to Real Analysis 5 The Integral | 69 |

A Concise Introduction to Real Analysis 6 Product Measures and Fubinis Theorem | 103 |

A Concise Introduction to Real Analysis 7 Functions of a Real Variable | 123 |

A Concise Introduction to Real Analysis 8 General Countably Additive Set Functions | 151 |

A Concise Introduction to Real Analysis 9 Examples of Dual Spaces from Measure Theory | 165 |

A Concise Introduction to Real Analysis 10 Translation Invariance in Real Analysis | 195 |

A Concise Introduction to Real Analysis Appendix The BanachTarski Theorem | 225 |

229 | |

231 | |